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Computing with Real Numbers  I. The LFT Approach to Real Number Computation  II. A Domain Framework for Computational Geometry
 PROC APPSEM SUMMER SCHOOL IN PORTUGAL
, 2002
"... We introduce, in Part I, a number representation suitable for exact real number computation, consisting of an exponent and a mantissa, which is an in nite stream of signed digits, based on the interval [ 1; 1]. Numerical operations are implemented in terms of linear fractional transformations ( ..."
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Cited by 16 (1 self)
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We introduce, in Part I, a number representation suitable for exact real number computation, consisting of an exponent and a mantissa, which is an in nite stream of signed digits, based on the interval [ 1; 1]. Numerical operations are implemented in terms of linear fractional transformations (LFT's). We derive lower and upper bounds for the number of argument digits that are needed to obtain a desired number of result digits of a computation, which imply that the complexity of LFT application is that of multiplying nbit integers. In Part II, we present an accessible account of a domaintheoretic approach to computational geometry and solid modelling which provides a datatype for designing robust geometric algorithms, illustrated here by the convex hull algorithm.
Computability of Partial Delaunay Triangulation and Voronoi Diagram (Extended Abstract)
, 2002
"... Using the domaintheoretic model for geometric computation, we define the partial Delaunay triangulation and the partial Voronoi diagram of N partial points in R² and show that these operations are domaintheoretically computable and effectively computable with respect to Hausdorff distance and Lebe ..."
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Cited by 6 (3 self)
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Using the domaintheoretic model for geometric computation, we define the partial Delaunay triangulation and the partial Voronoi diagram of N partial points in R² and show that these operations are domaintheoretically computable and effectively computable with respect to Hausdorff distance and Lebesgue measure. These results are obtained by showing that the map which sends three partial points to the partial disc passing through them is computable. This framework supports the design of robust algorithms for computing the Delaunay triangulation and the Voronoi diagram with imprecise input.
Basic Algorithms of Computational Geometry with Imprecise Input
, 2005
"... The domaintheoretic model of computational geometry provides us with continuous and computable predicates and binary operations. It can also be used to generalise the theory of computability for real numbers and real functions into geometric objects and geometric operations. A geometric object is c ..."
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The domaintheoretic model of computational geometry provides us with continuous and computable predicates and binary operations. It can also be used to generalise the theory of computability for real numbers and real functions into geometric objects and geometric operations. A geometric object is computable if it is the effective limit of a sequence of finitary partial objects of the same type as the original object. We are also provided with two different quantitative measures for approximation using the Hausdorff metric and the Lebesgue measure. In this thesis, we introduce a new data type to capture imprecise data or approximate points on the plane, given in the shape of compact convex polygons. This data type in particular includes rectangular approximation and is invariant under linear transformations of coordinate system. Based on the new data type, we define the notion of a number of partial geometric operations, including partial perpendicular bisector and partial disc and we show that these operations and the convex hull, Delaunay triangulation and Voronoi diagram are Hausdorff and Scott continuous and nestedly Hausdorff and Lebesgue computable. We develop algorithms to obtain the partial convex hull, partial Delaunay triangulation and partial Voronoi diagram. We prove that the complexity of the partial convex hull is N log N in 2D and 3D, whereas the partial Delaunay triangulation and partial Voronoi diagram algorithms for nondegenerate data have the same complexity as their classical counterparts. 2
Computability in Computational Geometry
"... Abstract. We promote the concept of object directed computability in computational geometry in order to faithfully generalise the wellestablished theory of computability for real numbers and real functions. In object directed computability, a geometric object is computable if it is the effective lim ..."
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Abstract. We promote the concept of object directed computability in computational geometry in order to faithfully generalise the wellestablished theory of computability for real numbers and real functions. In object directed computability, a geometric object is computable if it is the effective limit of a sequence of finitary objects of the same type as the original object, thus allowing a quantitative measure for the approximation. The domaintheoretic model of computational geometry provides such an object directed theory, which supports two such quantitative measures, one based on the Hausdorff metric and one on the Lebesgue measure. With respect to a new data type for the Euclidean space, given by its nonempty compact and convex subsets, we show that the convex hull, Voronoi diagram and Delaunay triangulation are Hausdorff and Lebesgue computable.
Visual Hull from Imprecise Polyhedral Scene
"... Abstract—We present a framework to compute the visual hull of a polyhedral scene, in which the vertices of the polyhedra are given with some imprecision. Two kinds of visual event surfaces, namely VE and EEE surfaces are modelled under the geometric framework to derive their counterpart object, name ..."
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Abstract—We present a framework to compute the visual hull of a polyhedral scene, in which the vertices of the polyhedra are given with some imprecision. Two kinds of visual event surfaces, namely VE and EEE surfaces are modelled under the geometric framework to derive their counterpart object, namely partial VE and partial EEE surfaces, which contain the exact information of all possible visual event surfaces given the imprecision in the input. Correspondingly, a new definition of visual number is proposed to label the cells of Euclidean space partitioned by partial VE and partial EEE surfaces. The overall algorithm maintains the same computational complexity as the classical method and generates a partial visual hull which converges to the classical visual hull as the input converges to an exact value. Keywordsvisual hull; shape from silhouettes; imprecise input; solid domain; quadratic surface I.
simple polygons. See Figure 1.
"... We pose the problem of constructing the possible hull of a set of n imprecise points: the union of convex hulls of all sets of n points, where each point is constrained to lie within a particular region of the plane. We give an optimal algorithm for the case when n = 2, and the regions are a point a ..."
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We pose the problem of constructing the possible hull of a set of n imprecise points: the union of convex hulls of all sets of n points, where each point is constrained to lie within a particular region of the plane. We give an optimal algorithm for the case when n = 2, and the regions are a point and a simple (possibly nonconvex) polygon. We then describe how the algorithm leads to an optimal algorithm for the case when n ≥ 2, and each region is a simple polygon. 1 present an algorithm for constructing the possible hull of a point and a simple (possibly nonconvex) polygon, and describe how this algorithm can be used as a subroutine to construct the possible hull of two or more
A Domain Theoretic Approach to Computational Geometry
, 2003
"... In computing applications such as CAD/CAM, that involve modelling of real world objects, a wide range of commonly used and generally accepted algorithms has been developed. Many of them, however, are based on invalid assumptions about the capability of the machines that implement them. ..."
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In computing applications such as CAD/CAM, that involve modelling of real world objects, a wide range of commonly used and generally accepted algorithms has been developed. Many of them, however, are based on invalid assumptions about the capability of the machines that implement them.
Part I Case for Support Computing with arbitrary precision curves
, 2004
"... Previous research track record and plans for the future The proposer has a longstanding interest in the theory and applications of computing with arbitrary precision (AP), which began with his PhD training (1996–2000) at the University of Birmingham under the supervision of Prof. Jung. PhD thesis. ..."
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Previous research track record and plans for the future The proposer has a longstanding interest in the theory and applications of computing with arbitrary precision (AP), which began with his PhD training (1996–2000) at the University of Birmingham under the supervision of Prof. Jung. PhD thesis. The thesis (Konečn´y 2000) built on the work of Wiedmer, Weihrauch, Edalat, Ko and others on representing real numbers as infinite streams of digits in an arbitrary precision computation. The thesis addressed, in this context, one of the first fundamental questions in computational complexity theory: Which realnumber functions can be computed to an arbitrary precision without an ever growing need for more memory? This question was answered for many different ways of representing the real numbers as infinite streams of symbols and for all reasonably wellbehaved functions (in some precise sense). This result is also described and proved in two journal articles (Konečn´y 2004, Konečn´y 2002), each for different types of realnumber representations. The thesis extends the articles in terms of the scope of representations and also generalises the theorem in another direction. It covers not only functions but also onetomany mappings. Such mappings arise naturally from the fact that each real number can be represented in many ways: for different representations of the same arguments different, but all correct results may be computed. More importantly, some frequently occurring practical problems, such as finding a zero of a polynomial, cannot be computed as a function but rather as a onetomany mapping. Work in Edinburgh. More recently, the proposer worked as a research fellow for the EPSRC funded project “Type